Actually the whole idea of this problem was to prove exactly your identity. My idea was to prove that zeroes and poles are the same. For poles I was able to prove that, but for the zeroes I needed help. So I posted this thread, and now you tell me to prove the identity, which I really don't know how .

Actually the whole idea of this problem was to prove exactly your identity. My idea was to prove that zeroes and poles are the same. For poles I was able to prove that, but for the zeroes I needed help. So I posted this thread, and now you tell me to prove the identity, which I really don't know how .

a) The function is a polynomial of degree , and it is simple to check that it satisfies (each time, only one term is on-zero). Thus is a degree polynomial with roots, hence it is zero.

Then I looked for a more explicit proof. It turns out to have calculus flavour.

c) can be decomposed into simple elements (I'm not sure of the english terminology): (simple poles). And you note that from this expression. Using this formula in gives , which are the coefficients of , hence .

I need to show carefully that the limit exists and yields the integral representation of the Gamma function.
This seems somehow trivial to me since as and the right side is just the definition of the Gamma function. Or am I wrong?

I need to show carefully that the limit exists and yields the integral representation of the Gamma function.
This seems somehow trivial to me since as and the right side is just the definition of the Gamma function. Or am I wrong?

You also have to justify why the left hand side converges to . You'll probably need the dominated convergence theorem for that. Use , and write to be in the setting of the dominated convergence theorem.